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Albert and Mahalel 1 EVALUATION OF A TOLL PAYMENT: A Game-Theory Approach Dr. Gila Albert (Corresponding author) Lecturer Department of Technology Management Holon Institute of Technology – H.I.T. Holon 58102, Israel Tel: 972 3 5026746 Fax: 972 3 5026650 Email: [email protected] Dr. David Mahalel Associate Professor Head of Transportation Research Institute Department of Civil and Environmental Engineering Technion - Israel Institute of Technology Technion city, Haifa 32000, Israel Tel./fax: 972 4 8292378 Email: [email protected] Albert and Mahalel 2 Abstract This paper focuses on the evaluation of a toll payment on one road of a congested system. A game-theory model is suggested to analyze the interaction between, and the decisions reached by, the parties involved in such a system: the users and the initiator who built and operates the road. The ability of the authority responsible for managing the system to influence the players’ decisions is analyzed. The main emphasis is the development of suitable tools to estimate the user-utility function. The model may provide more insight into the decision-making process and predict the players’ rational behavior, along with proposing means to attain a better equilibrium. Introduction Tolls have for many years been known to be among the most promising means available to transportation planners for influencing travel demand and relieving traffic congestion (Ibanez, 1992; Button, 1995; TRB, 1994; Johansson and Mattsson, 1995). Initially, tolls were used mainly to levy taxes for general purpose; later, they were often employed to finance transport infrastructure. They were also involved in roadprivatization schemes, the toll payment enabling implementation of the investment refund. Accordingly, the toll is set to maximize the initiator’s profit (Fielding and Klein, 1993; Harrop, 1993; Gittings, 1987). Albert and Mahalel 3 The effect of the toll payment on road-privatization schemes depends on the decisions reached by the parties involved in the transportation system. We can point out three major parties: the initiator who built and operates the road, the system’s users, and the authority responsible for managing the system. While the effect of the toll on the initiator is analyzed in terms of the profit gained by levying a toll, its effect on users is evaluated by their travel behavior. In the short run, the users might primarily change their route choice, mode of travel, time of travel, and adjust destinations (Ibanez, 1992; TRB, 1994). The toll fee is perceived as an “out-of-pocket cost,” which should present the users with better travel conditions, implying shorter travel time (Albert and Mahalel, 2006). In times of congestion, however, even this expense is unable to assure the driver a specific travel time and, consequently, a specific utility level; traffic congestion still exists, as road pricing might only alleviate but not eliminate it. In other words, during the period of congestion, externalities are introduced, causing the traffic-flow regime to be unstable and dependent upon travel decisions made by a marginal user. Therefore, travel time is very sensitive even to the slightest change in traffic flow (Gartner et al., 1995; McDonald et al., 1999). Since travel time is the most important component in transport-user utility, the user has difficulty in estimating this utility, which might affect his/her willingness to pay the toll fee for driving a passenger car during the congestion period. A review of the literature reveals that most studies and models in the area refer to the toll payment as a travel-cost component (for example, Ramjerdi, 1995; Verhoef et al, 1996). This presentation is not able to depict the users’ decision in this complex situation. As a result of the toll implementation, the initiator’s profit is affected by the users’ decisions, and the users’ utility is affected by the initiator’s decision about the Albert and Mahalel 4 toll fee. This paper’s aim is to evaluate the interaction between, and the decisions reached by the users and the initiator who built and operates the road in a congested road system. Because of the reciprocal effects that inhere in such interaction, a gametheory model is suggested to make this evaluation. The authority responsible for managing wishes to reduce the negative effects of the congestion. The authority can influence players’ choices in this model, and the need for its intervention is analyzed. One of the main emphases in this paper is the development of suitable tools to estimate the user-utility function, especially when a toll is imposed. The rest of this paper is organized as follows: The following sub-section reviews applications of game theory to transportation. The next section presents and describes the model, focusing on the user-utility function. Then the use of the proposed model is illustrated in an example. Finally, we summarize and draw conclusions. Applications of Non–Cooperative Game Theory to Transportation The potential applications of Non-Cooperative Game Theory to transportation were mentioned back in the 1970s. Congestion games, introduced by Rosenthal (1973), are applicable in describing a congested transportation system. Congestion games are non-cooperative games, in which the utility of a player from choosing an alternative (from a finite set of alternatives) largely depends on the number of other players choosing the same alternative. Fisk (1984) describes behavioral models from game theory that can be applied for planning and operating transportation systems. Studies conducted mostly in recent Albert and Mahalel 5 years deal with various applications of game theory models to different problems in transportation. Several of these are mentioned below. Bjrnskau and Elvik (1992) present a game-theory model that argues the traditional analysis of the capability of expected utility theory (EUT) in describing road users’ adaptation to law enforcement. Harker and Hong (1994) present a model of an internal market for railroad-track resources as an N-player non-cooperative game. Chen and Ben-Akiva (1998) integrate the dynamic traffic-control problem and the dynamic traffic-assignment problem as a non-cooperative game between a traffic authority and highway users with the aim of finding a mutually consistent, dynamic, system-optimal signal setting and dynamic, user-optimal traffic flow. Kita (1999) develops a two-person, non-zero-sum, non-cooperative game to describe the traffic behavior of a pair of merging and through cars, while explicitly considering the interaction between them. Bell (2000) describes a two-player non-cooperative game between the network user, who seeks a path to minimize the expected trip cost, and an “evil entity”, that chooses link-performance scenarios to maximize the expected trip cost. The Nash equilibrium reached in this game measures network performance when users are extremely pessimistic about the state of the network; it may therefore be used as the basis for a cautious approach to network design. The problems of analyzing travel behavior and the impact of tolls have also been addressed through non-cooperative-game theoretical models. Van Vugt (1995) uses the legendary “social dilemma” (also known as the “prisoner’s dilemma” or “the tragedy of the commons”) to study travel behavior in regard to the journey to work, and having to choose between a passenger car (drive alone) and public transport or car pool. James (1998) points out the potential of non-cooperative game theory as a tool to analyze the demand for passenger-car usage and illustrates the mutual effects Albert and Mahalel 6 among users through the well-known “chicken” game. Levinson (1999) focuses on revenue policies and toll rates that emerge at jurisdiction boundaries under alternative behaviors in the absence of congestion. Levinson considers the welfare implications of tolling at a frontier under alternative behavioral assumptions: different objectives (welfare maximizing, profit maximizing, cost recovery), willingness to cooperate on setting tolls, and different time frames (one-time interactions and repeated interactions). In a later study, Levinson (2005) develops congestion theory and congestion pricing theory from its micro-foundations: the interaction of two or more vehicles. Using game theory, with a two-player game, the emergence of congestion is shown to depend on the players’ relative evaluations of early arrival, late arrival, and journey delay. Joksimovic et al. (2005) use game theory to formulate and solve optimal tolls, with a focus on the road authority’s different policy objectives. The problem of determining optimal tolls is defined using utility maximization theory, including elastic demand on the travelers’ side and different objectives for the road authority. Game-theory notions are adopted in regard to different games, as well as different players, rules, and outcomes of the games played between travelers, on the one hand, and the road authority, on the other. Furthermore, it should be noted that the robust, widespread concept of “Nash equilibrium” (Nash, 1950) used for non-cooperative games coincides with the common “user equilibrium” (UE) concept, also known as “Wardrop’s first principle” or “Wardrop‘s equilibrium” (Wardrop, 1952), used for traffic assignment. These two concepts were developed separately in the 1950s, but their implication is identical. That is, under non-cooperative situation, a stable condition, i.e., equilibrium, is reached only when all players (drivers) adopt the best responses to correct beliefs; Albert and Mahalel 7 consequently, no player (driver) can increase utility (reduce travel time) by unilaterally changing strategy (e.g., choosing another route). Model Description Overview The proposed model deals with a congested transport system in which a toll is imposed on one route of the system connecting origin A to destination B. The willingness to pay toll for driving along this route is known. The toll is operated by an initiator, who sets the toll level in order to maximize revenue. The driver’s aim is to maximize utility from traveling from A to B. The users can choose between two alternatives: 1. To travel by passenger car and pay the respective toll (drive alone). 2. To use public transportation. It should be noted that the user’s set of alternatives could be extended to include other alternatives (e.g., car pool, a free alterative route); however, to simplify the analysis, we will deal with public transportation as the only alternative. In the model, public transportation properties, such as travel time and cost, are assumed to be constant, since public transportation does not share the same infrastructure (e.g., rail or rapid transit bus). No doubt that the utility of public transportation is also affected by the number of users choosing this alternative; however, the effect of the marginal user is lower, compared to his or her effect while using a passenger car. Albert and Mahalel 8 The initiator’s utility is affected by the users’ choices and willingness to pay the fee for the tolled route, while the users’ utility is in turn affected by the initiator’s decision about a toll fee. Non-cooperative Game theory can provide a framework for modeling the decision-making processes in this situation. Because of the reciprocal effects that inhere in such interaction, a strategic form game between the users and the initiator is established, as we assume complete information and common knowledge. Their strategy sets revolve around various toll fees. The authority that is responsible for managing a congested transportation system wishes to reduce its negative effects. Our analysis relates to the need for intervention. The authority is able to control public transport utility levels by several means: e.g., increasing public transport frequency. Because public transport utility is a major variable in the model, the authority is able to influence both the users’ and the initiator’s decisions in order to attain better system performance. Notations, Definitions, and Assumptions The following will be considered: N: The total number of potential tolled road users. i : Dichotomy variable, representing the choice made by user i (i=1,..N) when the toll imposed is s (s>0) 1 passenger car is chosen i = 0 else (public transportation is chosen) Albert and Mahalel 9 m( ) s : Vector, representing the choices made by all potential tolled road users when the toll imposed is s (s>0) m( ) s ( 1 ,..., i ,..., N ) m()s : Scalar, representing the number of road users who choose the passenger car when the toll imposed is s (s>0) N m( ) s i i 1 t[m()s] : The travel time along the route when the toll imposed is s (s>0). This variable is affected directly by the number of route users; the general relationship between travel time and traffic volume is assumed to be as represented in Figure 1. Player 1: A road user. U1[] : Player 1’s utility from choosing a passenger car; the formulation of this utility will be determined latter. K : Player 1’s utility from choosing public transportation _ x : Vector, representing the range of toll fees that player 1 is willing to pay to drive a passenger car on the toll road; player 1’s set of alternatives: _ x ( x1 ,.. xm ) _ x x : A toll fee that player 1 is willing to pay, a strategy of the user’s strategy set. Player 2: An initiator who operates and levies the toll. Albert and Mahalel 10 U2[] : Player 2’s utility from levying the toll; the formulation of this utility will be determined later. _ S : Vector, representing the range of toll fees that player 2 can levy; this is player 2’s set of alternatives. _ S ( S1 ,.. S J ) . _ s S : A toll that player 2 levies, a strategy of the initiator’s strategy set. {x*,s*}: a strategy profile that is a Nash equilibrium. Player 2 Utility Function The utility function of player 2, the initiator, is defined as the profit gained from imposing a toll on passenger-car driving along the route from A to B. In order to simplify the analysis, we will refer only to the revenue, and not to the costs. The revenue is equal to the initiator’s inflows from imposing a toll and, therefore, is equal to the number of road users along the tolled road multiplied by the toll fee: u2 = m()s s In line with travel demand analysis, if we assume that all the other variables that influence the number of road users (e.g., operation costs, convenience) are given, we can explore the effect of the toll fee on the number of road users. The initiator will set the toll at a level that implies zero marginal revenue. Player 1 Utility Function Albert and Mahalel 11 As mentioned previously, the utility of a user from driving a passenger car (whether a toll is imposed or not) along a specific route in times of congestion largely depends on the number of other users using the same route. The user’s utility function is negatively related to the number of other users; therefore: du1 0 dm S This function can be either convex or concave. That is: du12 0 d 2 m S or: du12 0 d 2 m S In addition, the toll is also related to player 1’s utility. Choosing a strategy (i.e., the toll fee the user is willing to pay) determines whether one uses public transportation or the passenger car. That is, if x<s, the toll fee that player 1 is willing to pay is lower than the toll that player 2 imposes. In this case, player 1 chooses public transportation, and therefore the utility attained is K. On the other hand, if x≥s, the toll fee that player 1 is willing to pay is equal to or higher than the toll that player 2 imposes. In this case, player 1 chooses the passenger car. If player 1 does bear the toll expense (i.e., an “outof-pocket cost” the user’s utility consists of three components, defined as follows: u1 = u1 [m()s] + u1 (x-s) - {u1 [m()x] - u1 [m()s] } Where: Albert and Mahalel 12 u1 [m()s] : The component of player 1’s utility that is affected directly by the other number of drivers using the same route if toll s is imposed. u1 (x-s): The component of player 1’s utility that is affected by the “difference” between the fee the driver is willing to pay and the fee actually paid. the component of player 1’s utility (note: f(0)=0). This component is calculated by multiplying the amount of currency saved by the value of a unit of currency saved. The value can be estimating as the average of changes in marginal utilities caused by a given change in the toll fee (an instance of this calculation is provided in the example section). u1 [m()x] : The component of player 1’s utility that is affected directly by the other number of drivers using the same route if toll x is imposed. Note that u1 can be negative and that its value depends on ratios among its components. In Israeli companies, employers tend to pay the operational costs, parking fees, and tolls of management-level employees’ cars and often also equip these employees with a company car. No doubt, in regard to our study, the travel behavior of such road users is completely different, since these users do not perceive the toll as an “out of pocket cost.” Therefore, the utility function will consist only of the first component, u1 [m()s]. Our analysis will consider this situation, as well. The “simple approach” is to analyze these users’ behavior; the “complicated approach” is to analyze the behavior of users who pay an “out-of- pocket cost.” Types of Player 1 Utility Function Albert and Mahalel 13 The components of player 1’s utility function, which refers to the impact of the other number of drivers using the same route if toll s is imposed (e.g., the first and the third components in the complicated approach), may differ. This depends largely on player 1’s sensitivity to congestion. That is, different players may perceive different utilities from similar travel times; as stated by Levinson (1999): “Congestion pricing is most meaningful when demand is heterogeneous, that is, different travelers have different values of time and differ in their disutility from congestion”. Accordingly, we will determine three types of users who differ in sensitivity to congestion. The utility function in the complicated approach is as follows (an extension to more types might be trivial): 1. High sensitivity to congestion (HS type): u1 1 m s 1 1 f ( x s) if x≥s m x m s K else HS type can represent driver with relatively high value of time. 2. Medium sensitivity to congestion (MS type): u1 1 1 1 f ( x s) if x≥s m( ) s m( ) s m( ) x K MS type can represent driver with average value of time. else Albert and Mahalel 14 3. Low sensitivity to congestion (LS type): u1 1 1 1 f ( x s) if x≥s lg[ m( ) s ] lg[ m x ] lg[ m s ] K else LS type can represent driver with relatively low value of time. The utility function in the simple approach is as follows (an extension to more types might be trivial in this case, too): 1. High sensitivity to congestion (HS type): u1 1 if x≥s m s K else 2. Medium sensitivity to congestion (MS type): 1 u1 m( ) s K if x≥s else 3. Low sensitivity to congestion (LS type): u1 1 lg[ m( ) s ] K if x≥s else Albert and Mahalel 15 As the user type becomes more sensitive to congestion, these components will gain lower utility for a given number of users. The user type is affected by his/her personal characteristics, which might include the user’s value of time, flexibility in work hours, etc. The Payoff Matrix The payoff matrix presents the payoffs (utilities) of both players in every possible strategy profile of the game. The general structure of the payoff matrix in our model is illustrated in Figure 2. As is common in Non-Cooperative Game Theory, player 1 plays on rows, and player 2 on columns. In each cell, the left value represents player 1’s utility, and the right value player 2s’. Nash Equilibrium In this game, a strategy profile {x*, s*} is a pure Nash equilibrium if both the following conditions hold: (i) u1(x*,s*) u1(x,s*) (ii) u2(x*,s*) u2(x*,s) x1 x xm s1 s sj That is, in Nash equilibrium, the utilities of both players are determined. The example presented in the next section will consider their transport and social applications. To simplify the analysis, we will discuss only a pure Nash equilibrium. Albert and Mahalel 16 Example The following example illustrates the manner in which the model is employed. Assuming the willingness to pay a toll for driving a passenger car along a certain route, is known, and a range of toll fees that the initiator is able to impose (NIS 2NIS10, at NIS 1 intervals). The example is analyzed according to the three types of player 1 sensitivity to congestion (HS, MS, and LS), and will consider both the simple and the complicated approach. Assuming a given data regarding the willingness to pay toll for driving alone along a specific route is as shown in Appendix 1. The utility of player 2 is calculated by multiplying the number of road users willing to pay a certain toll fee by the toll fee. The payoff matrixes for each of the player 1 sensitivities, using the simple approach, are presented in Appendix 2, Appendix 3, and Appendix 4, respectively. As can been seen, each of these three games has multiple pure Nash equilibriums. For example, if the user type is HS (Appendix 2), there are nine Nash equilibriums: {x=3,s=3}, {x=10, s=4}, {x=9,s=4}, {x=8,s=4}, {x=7,s=4}, {x=6,s=4}, {x=5,s=4}, {x=4, s=4}, and {x=2,s=4}. Note that in the first eight equilibriums listed, the alternative chosen is passenger car (since x≥s); in the last equilibrium listed, public transportation is chosen (since x<s). However, the equilibrium that is most likely to be reached also depends on the level of K, the utility derived from public transportation. Table 1, Table 2, and Table 3 summarize the model outcomes and point out the impact of the level of K on the choices made by the user and by the initiator. Obvious seems the result indicating that as user sensitivity to congestion increases, a Albert and Mahalel 17 lower level of utility from public transportation will be enough to persuade the driver to choose public transportation. An interesting result is that if the level of public transportation is relatively low, equilibrium can also be reached when the toll imposed by the initiator is lower. However, it should be noted that this equilibrium is a Pareto deficient equilibrium; that is, both players may attain higher utility in another equilibrium outcome, and therefore this Pareto deficient equilibrium is less likely to be reached. The payoff matrixes for the three player 1 types in the complicated approach are presented in Appendix 5, Appendix 6, and Appendix 7, respectively. As can be seen, each of these three games has two Nash equilibriums. In equilibrium, the toll imposed by the initiator is equal to NIS 4; the choices made by the user depend on the level of K, the utility derived from public transportation, as summarized in Table 4, Table 5, and Table 6. In line with the results of the simple approach, the complicated approach also indicates that as user sensitivity to congestion increases, a lower level of utility from public transportation will be enough to persuade the driver to choose public transportation. However, a comparison of the cut–off level of public transportation utility required to effect this switch of alternatives shows that a user paying an “out of pocket cost” will demand a higher level of utility (e.g., for the LS type, K=0.398 in the simple approach, compared to 0.454 in the complicated approach). On the one hand, this result is not intuitive; we may suspect that basically a user who does not pay the toll expense will demand a higher level of utility of public transportation to switch alternatives. On the other hand, it seems that a user paying an “out of pocket cost” has solid expectations that the toll payment will enable him or her to drive in Albert and Mahalel 18 good traffic conditions, and therefore this user strongly opts for the passenger car over public transportation. The authority that is responsible for managing a transportation system can influence both the users’ and the initiator’s decisions in order to attain a better equilibrium. In the model, the authority is able to control one of the major variables the public transport utility level by, e.g., changing public transport frequency, travel fares, etc. In such games, characterized by multiple Nash equilibrium, its intervention can determine which equilibrium is reached. Summary and Conclusion This paper describes a game-theory approach to evaluate the impact of a toll payment on one segment of a congested transportation system. Its aim is to provide more insight into evaluating the decisions reached by the parties involved in a congested road system: the users, the initiator who built and operates the road, and the authority responsible for managing the system. Because of the reciprocal effects between the users and the initiator that inhere in such interaction, a game-theory model was thought to be suitable to attain this evaluation. The effect of the toll on users was evaluated by the alternative chosen (passenger car or public transportation) and by the willingness to pay a fee for using a certain road when driving a passenger car. For that purpose, the main emphasis here was the development of suitable tools to estimate the user-utility function, especially when a toll is imposed. In order to explore the effect of varied drivers, three types of Albert and Mahalel 19 functions describing the user’s sensitivity to congestion, which may influence the decision process and the game outcome, and two classes of user types, were analyzed. As shown by the example, as user sensitivity to congestion increases, a lower level of utility from public transportation will be enough to persuade the driver to choose public transportation. Another result indicated that a user paying an “out-of-pocket cost” would demand a higher level of utility of public transportation in order to switch from passenger car to public transportation. The evaluation of the toll and its effect on the initiator was analyzed in terms of the revenue gained by levying a toll. The initiator’s utility is affected by the users’ choices, while the users’ utility is in turn affected by the initiator’s decision about a toll fee. As was illustrated, the user type has an impact on the toll levied by the initiator. Employing variable user types which have may make the problem more interesting. The existences of multiple Nash equilibriums point to the need for the interference of the authority responsible for managing the transportation system. It is well known that Nash equilibrium may not attain an efficient outcome. Nonetheless, by controlling the utility of public transportation, the authority may impact the equilibrium attained, the users’ utility, and the initiator’s revenue. The model described may capture the effect of changes in the system infrastructure, such as increasing road capacity, and in policy changes, such as taxation, on the decisionmaking process. This paper provided a framework for analyzing the complex relationships that exist in the transportation system, with the use of game theory. The level of information provided to the users and the initiator may affect the game structure. A dynamic game can represent a situation in which the initiator is the “leader.” Repeated Albert and Mahalel 20 games can incorporate phenomena that are important but not grasped when we restrict our attention to static, one-shot games. Travel behavior, we believe, can best be described as a repeated game, and this should be further investigated. References 1. Albert, G. and Mahalel, D. (2006) ‘The Demand Curve and Its Implication for Road Pricing’. Paper presented at the 3rd International Kuhmo Conference and Nectar 2 Meeting on "Pricing, Financing and Investment in Transport", July 11-14, 2006, Helsinki, Finland (Available on CD). 2. Bell, M.G.H. (2000) ‘A Game Theory Approach to Measuring the Performance Reliability of Transport Networks’, Transportation Research B34 (6), pp. 533545. 3. Bjrnskau, T. and Elvik, R. (1992) ‘Can Road Traffic Law Enforcement Permanently Reduce the Number of Accidents?’ Accident Analysis and Prevention 24, pp. 507-520. 4. Button, K. (1995) ‘Road Pricing as an Instrument in Traffic Management’, in: Road Pricing: Theory, Empirical Assessment and Policy. Edited by Johansson B., and Mattsson L.G, Kluwer Academic Publishers, pp. 35-56. 5. Chen, O. and Ben-Akiva, M. (1998) ‘Game-Theoretic Formulation of the Interaction between Dynamic Traffic Control and Dynamic Traffic Assignment’, Transportation Research Record 1617, pp. 179-188. 6. Committee for Study on Urban Transportation Congestion Pricing (1994). Curbing Gridlock: Peak Period Fees to Relieve Traffic Congestion. Special Albert and Mahalel 21 Report 242, Vol. 1. Transportation Research Board, National Academy Press, Washington, DC. 7. Fielding, G. J., and Klein D. B. (1993). ‘How to Franchise Highways’, Journal of Transport Economics and Policy 27, pp. 113-129. 8. Fisk C. S. (1984). ‘Game Theory and Transportation Systems Modeling’, Transportation Research B 18, pp. 301-313. 9. Gartner, N. H., Carroll, J. M., and Ajay, K. R. (1995). Monograph on Traffic Flow Theory, Transportation Research Board, Washington, DC. 10. Gittings, G.L. (1987). ‘Some Financial, Economic and Social Policy Issues Associated with Toll Finance’, Transportation Research Record 1107, pp. 20-30. 11. Harker, P.T. and Hong, S. (1994). ‘Pricing of Track Time in Railroad Operation: An International Market Approach’, Transportation Research, 28B (3), pp.197212. 12. Harrop, P. (1993). Charging for Road Use Worldwide. A Financial Times Management Report, London. 13. Ibanez, G. (1992). The Political Economy of Highway Tolls and Congestion Pricing. Transportation Quarterly 46, pp. 343-360. 14. James, T. (1998). ‘A Game Theoretic Model of Road Usage’, Mathematics in Transport Planning and Control, Proceedings of the Third IMA International Conference on Mathematics in Transport Planning and Control, pp. 401-409. 15. Johansson, B., and Mattsson, L.G. (1995) ‘Principles of Road Pricing’, in: Road Pricing: Theory, Empirical Assessment and Policy. Edited by Johansson, B., and Mattsson, L.G. Kluwer, Academic Publishers, pp. 7-34. Albert and Mahalel 22 16. Joksimovic, D., Bliemer, M., and Bovy, P.H.L. (2005). ‘Different Policy Objectives of the Road Pricing Problem with Elastic Demand – A Game Theory Approach’, Compendium of Papers of the 84th Annual Meeting, Transportation Research Board, Washington, DC. 17. Kita, H. (1999). ’A Merging- Giveaway Interaction Model of Cars in a Merging Section: A Game Theoretic Analysis’, Transportation Research 33A (3/4), pp. 305-312. 18. Levinson, D. (1999). ‘Tolling at a Frontier: A Game Theoretic Analysis’, Proceedings of the 14th International Symposium of Transportation and Traffic Theory, Jerusalem, pp. 665-682. 19. Levinson, D. (2005). ‘Micro-foundations of Congestion and Pricing: A Game Theory Perspective’, Transportation Research A 39, pp. 691-704. 20. McDonald, J. F., d’Ouville, E. L., and Liu, L. N. (1999). ‘Economics of Urban Highway Congestion and Pricing’, Transportation Research, Economics and Policy, Kluwer Academic Publishers. 21. Nash, J. F. (1950). ‘Equilibrium Points in n-person Games’, National Academy of Sciences 36, pp. 48-49. 22. Ramjerdi, F. (1995). ‘An Evaluation of the Impact of the Oslo Toll Scheme on Travel Behaviour’, in: Road Pricing and Toll Financing. Inst. of Transportation Economics, Norway, pp. 87-111. 23. Rosenthal, R.W. (1973). ‘A Class of Games Possessing Pure-Strategy Nash Equilibria’, Int. Journal Game Theory 2, pp. 65-67. 24. Van Vugt, M. (1995). Social Dilemmas and Transportation Decisions, University of Southampton. Albert and Mahalel 23 25. Verhoef, E., Nijkamp, P., and Rietveld, P. (1996). ‘Second-Best Congestion Pricing: The Case of an Untolled Alternative’, Journal of Urban Economics 40 (3), pp. 279-302. 26. Wardrop, J. G. (1952) ‘Some Theoretical Aspects of Road Traffic Research’, Proceedings of the Institute of Civil Engineers, Part II, Vol. 1, pp. 325-378. Albert and Mahalel 24 Appendix 1: The willingness to pay toll - data for the example Toll No. of drivers willing to pay 10 9 8 7 6 5 4 3 2 52 104 158 166 226 250 323 430 600 Albert and Mahalel 25 Appendix 2: Payoff matrix for Example 1, Simple Approach, type HS 10 9 8 7 6 5 4 3 2 10 0.02 k k k k k k k k 530 520 520 520 520 520 520 520 520 9 0.010 0.010 k k k k k k k 945 945 936 936 936 936 936 936 936 8 0.006 0.006 0.006 k k k k k k 1272 1272 1272 1264 1264 1264 1264 1264 1264 7 0.006 0.006 0.006 0.006 k k k k k 1169 1169 1169 1169 1162 1162 1162 1162 1162 6 0.004 0.004 0.004 0.004 0.004 k k k k 1362 1362 1362 1362 1362 1356 1356 1356 1356 5 0.004 0.004 0.004 0.004 0.004 0.004 k k k 1255 1255 1255 1255 1255 1255 1250 1250 1250 4 0.003 0.003 0.003 0.003 0.003 0.003 0.003 k k 1296 1296 1296 1296 1296 1296 1296 1292 1292 3 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 k 1293 1293 1293 1293 1293 1293 1293 1293 1290 2 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 0.002 1202 1202 1202 1202 1202 1202 1202 1202 1200 Albert and Mahalel 26 Appendix 3: Payoff matrix for Example 1, Simple Approach, type MS 10 9 8 7 6 5 4 3 2 10 0.14 k k k k k k k k 530 520 520 520 520 520 520 520 520 9 0.098 0.098 k k k k k k k 945 945 936 936 936 936 936 936 936 8 0.079 0.079 0.079 k k k k k k 1272 1272 1272 1264 1264 1264 1264 1264 1264 7 0.077 0.077 0.077 0.077 k k k k k 1169 1169 1169 1169 1162 1162 1162 1162 1162 6 0.066 0.066 0.066 0.066 0.066 k k k k 1362 1362 1362 1362 1362 1356 1356 1356 1356 5 0.063 0.063 0.063 0.063 0.063 0.063 k k k 1255 1255 1255 1255 1255 1255 1250 1250 1250 4 0.056 0.056 0.056 0.056 0.056 0.056 0.056 k k 1296 1296 1296 1296 1296 1296 1296 1292 1292 3 0.048 0.048 0.048 0.048 0.048 0.048 0.048 0.048 k 1293 1293 1293 1293 1293 1293 1293 1293 1290 2 0.041 0.041 0.041 0.041 0.041 0.041 0.041 0.041 0.041 1202 1202 1202 1202 1202 1202 1202 1202 1200 Albert and Mahalel 27 Appendix 4 : Payoff Matrix for Example 1, Simple Approach, type LS 10 9 8 7 6 5 4 3 2 10 0.58 k k k k k k k k 530 520 520 520 520 520 520 520 520 9 0.495 0.495 k k k k k k k 945 945 936 936 936 936 936 936 936 8 0.454 0.454 0.454 k k k k k k 1272 1272 1272 1264 1264 1264 1264 1264 1264 7 0.450 0.450 0.450 0.450 k k k k k 1169 1169 1169 1169 1162 1162 1162 1162 1162 6 0.424 0.424 0.424 0.424 0.424 k k k k 1362 1362 1362 1362 1362 1356 1356 1356 1356 5 0.417 0.417 0.417 0.417 0.417 0.417 k k k 1255 1255 1255 1255 1255 1255 1250 1250 1250 4 0.398 0.398 0.398 0.398 0.398 0.398 0.398 k k 1296 1296 1296 1296 1296 1296 1296 1292 1292 3 0.380 0.380 0.380 0.380 0.380 0.380 0.380 0.380 k 1293 1293 1293 1293 1293 1293 1293 1293 1290 2 0.360 0.360 0.360 0.360 0.360 0.360 0.360 0.360 0.360 1202 1202 1202 1202 1202 1202 1202 1202 1200 Albert and Mahalel 28 Appendix 5: Payoff matrix for Example 1, Complicated Approach, type HS 10 9 8 7 6 5 4 3 2 10 0.02 k k k k k k k k 530 520 520 520 520 520 520 520 520 9 0.002 0.010 k k k k k k k 945 945 936 936 936 936 936 936 936 8 0.00 0.005 0.006 k k k k k k 1272 1272 1272 1264 1264 1264 1264 1264 1264 7 0.000 0.007 0.008 0.006 k k k k k 1169 1169 1169 1169 1162 1162 1162 1162 1162 6 0.00 0.006 0.007 0.005 0.004 k k k k 1362 1362 1362 1362 1362 1356 1356 1356 1356 5 0.000 0.007 0.008 0.006 0.006 0.004 k k k 1255 1255 1255 1255 1255 1255 1250 1250 1250 4 0.000 0.007 0.008 0.007 0.006 0.004 0.003 k k 1296 1296 1296 1296 1296 1296 1296 1292 1292 3 0.001 0.008 0.009 0.007 0.007 0.005 0.004 0.002 k 1293 1293 1293 1293 1293 1293 1293 1293 1290 2 0.002 0.009 0.010 0.008 0.008 0.006 0.005 0.003 0.002 1202 1202 1202 1202 1202 1202 1202 1202 1200 Albert and Mahalel 29 Appendix 6: Payoff Matrix for Example 1, Complicated Approach, type MS 10 9 8 7 6 5 4 3 2 10 0.14 k k k k k k k k 530 520 520 520 520 520 520 520 520 9 0.070 0.098 k k k k k k k 945 945 936 936 936 936 936 936 936 8 0.05 0.073 0.079 k k k k k k 1272 1272 1272 1264 1264 1264 1264 1264 1264 7 0.05 0.081 0.087 0.077 k k k k k 1169 1169 1169 1169 1162 1162 1162 1162 1162 6 0.04 0.07 0.077 0.067 0.066 k k k k 1362 1362 1362 1362 1362 1356 1356 1356 1356 5 0.05 0.08 0.083 0.073 0.072 0.063 k k k 1255 1255 1255 1255 1255 1255 1250 1250 1250 4 0.05 0.07 0.08 0.070 0.069 0.060 0.056 k k 1296 1296 1296 1296 1296 1296 1296 1292 1292 3 0.04 0.07 0.08 0.067 0.066 0.057 0.053 0.048 k 1293 1293 1293 1293 1293 1293 1293 1293 1290 2 0.04 0.07 0.07 0.06 0.063 0.054 0.050 0.045 0.041 1202 1202 1202 1202 1202 1202 1202 1202 1200 Albert and Mahalel 30 Appendix 7: Payoff Matrix for Example 1, Complicated Approach, type LS 10 9 8 7 6 5 4 3 2 10 0.58 k k k k k k k k 530 520 520 520 520 520 520 520 520 9 0.438 0.495 k k k k k k k 945 945 936 936 936 936 936 936 936 8 0.38 0.442 0.454 k k k k k k 1272 1272 1272 1264 1264 1264 1264 1264 1264 7 0.40 0.461 0.474 0.450 k k k k k 1169 1169 1169 1169 1162 1162 1162 1162 1162 6 0.38 0.438 0.451 0.427 0.424 k k k k 1362 1362 1362 1362 1362 1356 1356 1356 1356 5 0.39 0.451 0.463 0.440 0.437 0.417 k k k 1255 1255 1255 1255 1255 1255 1250 1250 1250 4 0.38 0.442 0.454 0.431 0.428 0.408 0.398 k k 1296 1296 1296 1296 1296 1296 1296 1292 1292 3 0.375 0.432 0.445 0.421 0.419 0.398 0.389 0.380 k 1293 1293 1293 1293 1293 1293 1293 1293 1290 2 0.36 0.421 0.433 0.410 0.407 0.387 0.377 0.368 0.360 1202 1202 1202 1202 1202 1202 1202 1202 1200 Albert and Mahalel 31 Table 1: Equilibrium in the example, Simple Approach, type HS Alternative chosen by Toll imposed by Level of K Player 1 Player 2 (in NIS) Passenger car 4 or 3 K Passenger car 4 0.002<K<0.003 Public transportation 4 K>0.003 Note: If K=0.003, then player 1 is indifferent when choosing between passenger car and public transportation. Albert and Mahalel 32 Table 2: Equilibrium in the Example, Simple Approach, type MS Alternative chosen by Toll imposed by Level of K Player 1 Player 2 (in NIS) Passenger car 4 or 3 K48 Passenger car 4 0.048<K<0.056 Public transportation 4 K>0.056 Note: If K=0.056, player 1 is indifferent when choosing between passenger car and public transportation. Albert and Mahalel 33 Table 3: Equilibrium in the Example, Simple Approach, type LS Alternative chosen by Toll imposed by Level of K Player 1 Player 2 (in NIS) Passenger car 4 or 3 K380 Passenger car 4 0.380<K<0.398 Public transportation 4 K>0.398 Note: If K=0.398, player 1 is indifferent when choosing between passenger car and public transportation. Albert and Mahalel 34 Table 4: Equilibrium in the Example, Complicated Approach, type HS Alternative chosen by Toll imposed by Level of K Player 1 Player 2 (in NIS) Passenger car 4 K<008 Public transportation 4 K>0.008 Note: If K=0.003, player 1 is indifferent when choosing passenger car and public transportation. Albert and Mahalel 35 Table 5: Equilibrium in the Example, Complicated Approach, type MS Alternative chosen by Toll imposed by Player 1 Player 2 (in NIS) Passenger car 4 K<08 Public transportation 4 K>0.08 Note: If K=0.08, player 1 is indifferent when choosing between passenger car and public transportation. Level of K Albert and Mahalel 36 Table 6: Equilibrium in the Example, Complicated Approach, type LS Alternative chosen by Toll imposed by Level of K Player 1 Player 2 (in NIS) Passenger car 4 K<454 Public transportation 4 K>0.454 Note: If K=0.454, player 1 is indifferent when choosing between passenger car and public transportation. Albert and Mahalel 37 List of Figures Figure 1: Relationship between travel time and traffic volume…..37 Figure 2: The game payoff matrix………………………………...38 Albert and Mahalel 38 Travel time Figure 1: Relationship between travel time and traffic volume No. of drivers Albert and Mahalel 39 Figure 2: The game payoff matrix 1/2 s1 .. .. x1 u1(x1,s1) u2(x1,s1) u1(x1,sj) u2(x1,sj) x2 u1(x2,s1) u2(x2,s1) u1(x2,sj) u2(x2,sj) xm u1(xm,s1) u2(xm,s1) u1(xm,sj) u2(xm,sj) sj .. ..